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Magnetic Backgrounds and Tachyonic Instabilities in Closed String Theory

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Magnetic Backgrounds and Tachyonic Instabilities in Closed String Theory Magnetic backgrounds and tachyonic instabilities in closed string theory Cite as: AIP Conference Proceedings 607, 269 (2002); https://doi.org/10.1063/1.1454381 Published Online: 25 February 2002 A. A. Tseytlin AIP Conference Proceedings 607, 269 (2002); https://doi.org/10.1063/1.1454381 607, 269 © 2002 American Institute of Physics. Magnetic Backgrounds and Tachyonic Instabilities in Closed String Theory A.A. Tseytlin Department of Physics, The Ohio State University, Columbus OH 43210-1106, USA Abstract. We consider closed superstrings in Melvin-type magnetic backgrounds. In the case of NS-NS backgrounds, these are exactly solvable as weakly coupled string models with spectrum containing tachyonic modes. Magnetic field allows one to interpolate between free superstring the- ories with periodic and antiperiodic boundary conditions for the fermions around some compact direction, and, in particular, between type 0 and type II string theories. Using "9-11" flip, this inter- polation can be extended to M-theory and may be used to study the issue of tachyon condensation in type 0 string theory. We review related duality proposals, and, in particular, suggest a description of type 0 theory in terms of M-theory in a curved magnetic flux background in which the type 0 tachyon appears to correspond to a state in d=l 1 supergravity fluctuation spectrum. INTRODUCTION Magnetic backgrounds play an important role in field theory and open string theory pro- viding simple solvable models with nontrivial physics content. Similar (approximately) constant magnetic backgrounds in gravitational theories like closed string theory are necessarily curved with an example provided by the Melvin-type flux tube solutions (see, e.g., [1]). The role of the magnetic field(s) is played by the vector(s) originating from the metric and/or antisymmetric 2-tensor upon Kaluza-Klein reduction on a circle. One important special case is the Kaluza-Klein Melvin background which has curved magnetic universe interpretation in 9 dimensions, but, remarkably, is represented by a flat ("twisted") 10-dimensional space [2]. The non-trivial 3-dimensional part of this space is a twisted product of a K-K circle and a 2-plane: going around the circle must be accompanied by a rotation by angle 2nb in the plane. This is a continuous version of the Rohm model [3] where the 2-plane was replaced by a 2-torus so that the magnetic ("twist") parameter was allowed to take only discreet values. The flatness of the 10-d space allows one to solve the corresponding (super)string theory exactly [4, 5] determining the "Landau-type" spectrum of states which (for large enough magnetic field) contains tachyons in the winding sector. Other NS-NS Melvin- type models (which are no longer flat in 10 dimensions) are also solvable since they may be formally related to the K-K Melvin model by a generalized O(d,d) T-duality transformation [6]. One can also construct similar magnetic backgrounds with R-R magnetic field by applying U-duality transformation, or equivalently, by lifting, e.g., the K-K Melvin space to 11 dimensions and then reducing down to 10 along a different dimension [7, 8]. Such R-R string models seem, however, hard to solve exactly. CP607, String Theory, edited by H. Aoki and T. Tada © 2002 American Institute of Physics 0-7354-0051-2/027$ 19.00 269 One characteristic feature of these magnetic closed string models is that the magnetic field parameter introduces an effective phase for space-time fermions; in particular, switching on magnetic field allows one to interpolate between periodic and antiperiodic boundary conditions for the fermions along a spatial circle [4, 7]. Antiperiodic fermions appear in the context of superstrings at finite temperature [9] and in closely related type 0 string theory viewed as (—I)FS orbifold of type II superstring theory [10, 11]. Using the orbifold interpretation of type 0 string theory and "9-11" flip it was sug- gested in [12] to interpret type 0 string theory as a similar non-super symmetric orbifold compactification of 11-d M-theory. It was conjectured [12] that the tachyon of type OA string in flat space gets m2 > 0 at strong coupling where type OA string becomes dual to 11-d M-theory on large circle with antiperiodic fermions. In an interesting paper [13], this proposal was combined with the observation that K-K Melvin magnetic background in type II string theory (and its direct lift to 11 dimensions) may be used to interpolate [4, 7] between periodic and antiperiodic boundary conditions for space-time fermions. Since reducing the flat twisted 11-d space along a different - "mixed circle" - direction gives [2, 7] the R-R 7-brane background, it was suggested that type 0 string in this 10-d R-R Melvin background should be dual to type II theory in the same background but with shifted magnetic field. This implies that type 0 string tachyon should disappear at strong magnetic R-R field [13, 14]. In our recent work [8], which will be reviewed below, we extended the discussion in [7, 13] to the case of more general class of Melvin-type backgrounds with two independent magnetic parameters b and b (parametrizing the vector and axial vector 9- d fields originating from G® and B& components). This class includes the K-K Melvin (b ^ 0, b = 0) and the dilatonic Melvin (b — b) solutions as special cases and is covariant under T-duality. We lifted these type IIA magnetic solutions to 11 dimensions, getting a b ^ 0 generalization of the flat [2] d=l 1 background discussed in [7, 13]. Dimensional reduction along different directions (or U-dualities directly in d=10) led to a number of d=10 supergravity backgrounds with R-R magnetic fluxes, generalizing to b ^ 0 the R-R magnetic flux 7-brane [7, 13] of type IIA theory. We proposed the analog of the type II - type 0 duality in [13] now based on a curved d=l 1 background which is a lift of the T-dual b ^ 0, b — 0 Melvin background. Remark- ably, this background has manifest 9-11 symmetry, i.e. its two different 10-d reductions are formally the same NS-NS Melvin spaces. While in the original K-K Melvin set-up the tachyon originates from a stringy winding mode (i.e. type 0 tachyon may be inter- preted as corresponding to a wound membrane state) in this T-dual setting the tachyon appears to be a momentum mode, i.e. the type 0 tachyon appears to correspond to a state in d=l 1 supergravity multiple!. One motivation for the study of these magnetic backgrounds and their instabilities is to get better understanding of closed string tachyons and their stabilization, i.e. of dynamical interpolations between stable and unstable theories. In particular, one would like to establish precise relations between non-supersymmetric backgrounds in type II superstring theory and unstable backgrounds in type 0 theory. Another is to use interpolating magnetic backgrounds to connect D-brane solutions in the two theories and related gauge theories. Understanding the fate of closed string tachyon in such non- supersymmetric situations is crucial, in particular, for the program of constructing string theory duals of non-supersymmetric gauge theories [15, 16] (see, e.g., [17, 18, 19] for 270 some related work). NS-NS MELVIN BACKGROUND IN D^IO SUPERSTRING Consider the following NS-NS bosonic background in type II string theory: r2 (1) + bzr2 This is an exact solution of the string theory - the corresponding sigma model is 7 conformal to all orders in a [4, 5], x9 is a periodic coordinate of radius R = R9 and <p is the angular coordinate with period 2n. This model is covariant under T-duality in x9 direction: the (R,b,b) model is T-dual (x9 -» xg) to (R,b,b) model with R ~ ^. The constants b and b are the magnetic field parameters. The magnetic Melvin- type flux tube interpretation becomes apparent upon dimensional reduction in the x9- direction. The resulting solution of d = 9 supergravity (3) (4) (5) describes a magnetic flux tube universe with the magnetic vector A (coming from the metric) and the axial-vector B (coming from the 2-form) and the non-constant dilaton <|) and K-K scalar a. There are two special important cases: (i) b = Q,b ^ 0 and (ii) b = 0,£ ^ 0. The corresponding string backgrounds are T-dual to each other. The first (K-K Melvin model) is represented by the flat 10-d space (but curved magnetic 9-d background) 2 2 2 2 2 ds = -dt + dx] + dxl + dr + r (d($> + b dx9) . (6) The second has non-trivial metric, 3-form and dilaton: 2 2 2 2 l 2 2 2 ds w = -dt + dx + dr + f~ (dx 9 + r </cp ) • (7) 2 H3 = dB2 = 2bf~ rdr A dy A dx9 , ^(^-^o) = f- 1 _ (8) They define equivalent conformal field theories, with the two different geometries being "seen", as in other known T-dual situations, by point-like momentum and winding string modes respectively. * 1 While the curved 2-parameter background (1) may look much more complicated than the flat one (6), it is the natural T-duality (O(2,2) duality) "completion" of (6). A possible analogy is to think of (6) as a counterpart of a plane wave background; then (7) corresponds to the fundamental string background, and (1) - to a superposition of a fundamental string and a wave. 271 For generic values of b and b all of the type II supersymmetries are broken; supersym- metries are restored at the special values bR = 2n\ and bR — 2ri2 (HI = 0, ± 1,...), where the conformal model describes the standard flat-space type II superstring theory.
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